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500 Chapter 7: Transcendental Functions
EXERCISES 7.4
Algebraic Calculations With and 
Simplify the expressions in Exercises 1–4.
1. a. b. c.
d. e. f.
2. a. b. c.
d. e. f.
3. a. b. c.
4. a. b. c.
Express the ratios in Exercises 5 and 6 as ratios of natural logarithms
and simplify.
5. a. b. c.
6. a. b. c.
Solve the equations in Exercises 7–10 for x.
7.
8.
9.
10.
Derivatives
In Exercises 11–38, find the derivative of y with respect to the given
independent variable.
11. 12.
13. 14.
15. 16. y = t1- ey = xp
y = 2ss
2dy = 52s y = 3-xy = 2x
ln e + 4-2 log4 sxd = 1x log10 s100d
3log3 sx
2d = 5e ln x - 3 # 10log10 s2d
8log8 s3d - e ln 5 = x2 - 7log7 s3xd
3log3 s7d + 2log2 s5d = 5log5 sxd
log a b
log b a
log210 x
log22 xlog 9 xlog 3 x
log x a
log x2 a
log 2 x
log 8 x
log 2 x
log 3 x
log4 s2e
x sin xdloge sexd25log5 s3x
2d
log2 se sln 2dssin xdd9log3 x2log4 x
log3 a19blog121 11log11 121
plogp 710log10 s1>2d2log2 3
log4 a14blog323log4 16 1.3log1.3 758log8225log5 7
loga xax 17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.
37. 38.
Logarithmic Differentiation
In Exercises 39–46, use logarithmic differentiation to find the deriva-
tive of y with respect to the given independent variable.
39. 40.
41. 42.
43. 44.
45. 46.
Integration
Evaluate the integrals in Exercises 47–56.
47. 48. Ls1.3d
x dxL 5
x dx
y = sln xdln xy = xln x
y = xsin xy = ssin xdx
y = t2ty = s1tdt y = xsx+ 1dy = sx + 1dx
y = t log 3 Ae ssin tdsln 3d By = log 2 s8t ln 2d
y = 3 log8 slog2 tdy = 3log2 t
y = log2 a x2e2
22x + 1 by = log5 ex y = log7 a
sin u cos u
eu 2u
by = u sin slog7 ud
y = log5 B a 7x3x + 2 b ln 5y = log3 a ax + 1x - 1 b ln 3b y = log3 r # log9 ry = log2 r # log4 r
y = log25 ex - log51xy = log4 x + log4 x2 y = log3s1 + u ln 3dy = log2 5u y = 5-cos 2ty = 2sin 3t
y = 3tan u ln 3y = 7sec u ln 7
y = sln udpy = scos ud22
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7.4 and 501loga xa x
49. 50.
51. 52.
53. 54.
55. 56.
Evaluate the integrals in Exercises 57–60.
57. 58.
59. 60.
Evaluate the integrals in Exercises 61–70.
61. 62.
63. 64.
65. 66.
67. 68.
69. 70.
Evaluate the integrals in Exercises 71–74.
71. 72.
73. 74.
Theory and Applications
75. Find the area of the region between the curve 
and the interval of the x-axis.
76. Find the area of the region between the curve and the in-
terval of the x-axis.
77. Blood pH The pH of human blood normally falls between 7.37
and 7.44. Find the corresponding bounds for 
78. Brain fluid pH The cerebrospinal fluid in the brain has a hy-
dronium ion concentration of about 
per liter. What is the pH?
79. Audio amplifiers By what factor k do you have to multiply the
intensity of I of the sound from your audio amplifier to add 10 dB
to the sound level?
80. Audio amplifiers You multiplied the intensity of the sound of
your audio system by a factor of 10. By how many decibels did
this increase the sound level?
[H3 O
+] = 4.8 * 10-8 moles
[H3 O
+] .
-1 … x … 1
y = 21- x
-2 … x … 2
y = 2x>s1 + x2d
1
ln a
 L
x
1
 
1
t dt, x 7 0L
1>x
1
 
1
t dt, x 7 0
L
e x
1
 
1
t dtL
ln x
1
 
1
t dt, x 7 1
L 
dx
xslog8 xd2L 
dx
x log10 x
L
3
2
 
2 log2 sx - 1d
x - 1 dxL
9
0
 
2 log10 sx + 1d
x + 1 dx
L
10
1>10
 
log10 s10xd
x dxL
2
0
 
log2 sx + 2d
x + 2 dx
L
e
1
 
2 ln 10 log10 x
x dxL
4
1
 
ln 2 log2 x
x dx
L
4
1
 
log2 x
x dxL 
log10 x
x dx
L
e
1
 xsln 2d - 1 dxL
3
0
s12 + 1dx12 dx Lx22- 1 dxL 3x23 dx
L
2
1
 
2ln x
x dxL
4
2
 x2xs1 + ln xd dx
L
p>4
0
 a1
3
b tan t sec2 t dtL
p>2
0
 7cos t sin t dt
L
4
1
 
21x1x dxL221 x2sx2d dx L
0
-2
 5-u duL
1
0
 2-u du
81. In any solution, the product of the hydronium ion concentration
(moles L) and the hydroxyl ion concentration 
(moles L) is about 
a. What value of minimizes the sum of the concentrations,
(Hint: Change notation. Let
)
b. What is the pH of a solution in which S has this minimum
value?
c. What ratio of to minimizes S?
82. Could possibly equal Give reasons for your an-
swer.
83. The equation has three solutions: and one
other. Estimate the third solution as accurately as you can by
graphing.
84. Could possibly be the same as for Graph the two
functions and explain what you see.
85. The linearization of
a. Find the linearization of at Then round its
coefficients to two decimal places.
b. Graph the linearization and function together for
and 
86. The linearization of
a. Find the linearization of at Then round
its coefficients to two decimal places.
b. Graph the linearization and function together in the window
and 
Calculations with Other Bases
87. Most scientific calculators have keys for and ln x. To find
logarithms to other bases, we use the Equation (5), 
Find the following logarithms to five decimal places.
a. b.
c. d.
e. ln x, given that 
f. ln x, given that 
g. ln x, given that 
h. ln x, given that 
88. Conversion factors
a. Show that the equation for converting base 10 logarithms to
base 2 logarithms is
b. Show that the equation for converting base a logarithms to
base b logarithms is
logb x =
ln a
ln b
 loga x .
log2 x =
ln 10
ln 2
 log10 x .
log10 x = -0.7
log2 x = -1.5
log2 x = 1.4
log10 x = 2.3
log0.5 7log20 17
log7 0.5log3 8
sln xd>sln ad .
log a x =
log10 x
2 … x … 4.0 … x … 8
x = 3.ƒsxd = log3 x
log3 x
-1 … x … 1.-3 … x … 3
x = 0.ƒsxd = 2x
2x
x 7 0?2ln xxln 2
x = 2, x = 4,x2 = 2x
1>logb a?loga b
[OH-][H3 O
+]
x = [H3 O+] .
S = [H3 O+] + [OH-]?
[H3 O
+]
10-14 .> [OH
-]>[H3 O+]
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89. Orthogonal families of curves Prove that all curves in the family
(k any constant) are perpendicular to all curves in the family
(c any constant) at their points of intersection. (See
the accompanying figure.)
90. The inverse relation between and ln x Find out how good
your calculator is at evaluating the composites
91. A decimal representation of e Find e to as many decimal
places as your calculator allows by solving the equation ln x = 1.
eln x and ln sexd .
e x
y = ln x + c
y = - 1
2
 x2 + k
92. Which is bigger, or Calculators have taken some of the
mystery out of this once-challenging question. (Go ahead and
check; you will see that it is a surprisingly close call.) You can an-
swer the question without a calculator, though.
a. Find an equation for the line through the origin tangent to the
graph of 
b. Give an argument based on the graphs of and the
tangent line to explain why for all positive 
c. Show that for all positive 
d. Conclude that for all positive 
e. So which is bigger, or ep?pe
x Z e .xe 6 ex
x Z e .ln sxed 6 x
x Z e .ln x 6 x>e
y = ln x
[–3, 6] by [–3, 3]
y = ln x .
ep?pe
502 Chapter 7: Transcendental Functions
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